Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning ...

Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning and Precision FixturingSubmitted to Precision Engineering4.4. System stiffness modelingIn modeling the stiffness of the HPF, we assume that the groove flexure bearing, the guiding flexurebearing and the ball-groove contacts comprise the major sources of compliant deformations. As such,they are modeled as compliant elements, e.g. springs, using the direct stiffness approach. The flexurebearings are modeled as linear compliant elements and the bal-groove joints are modeled as non-linearcompliant Hertzian contacts. Each spring is assigned a local coordinate system. The compliance of theflexible parts (e.g. beams) within each element in the element’s local coordinate system is transformedinto reference frame of the element and the contributions of each flexible part are then combined to forma stiffness matrix for the element. The element’s stiffness characteristics are then transformed into theHPF’s coordinate system and then the contributions of each element are combined to form a stiffnessmatrix for the HPF. This matrix relates the motion of one coupled component (the ball-equippedcomponent) with respect to the grounded (groove-equipped component) under the influence of externalloads. This model was implemented in Matlab and used to predict the system stiffness as a function offixture preload. The change in stiffness with preload (due to the non-linear ball-groove contact stiffness)may be seen in Table 2.Table 2: Simulated HPF stiffness at coupling centroid – based on Matlab modelPreload Kxx[N/µm]Kyy [N/µm] Kzz [N/µm] Kxx[kNm/rad]Kyy[kNm/rad]Kzz[kNm/rad]450N 21.8 21.8 15.5 50.2 50.2 240.8225N 20.4 20.4 14.6 47.3 47.3 223.045N 16.2 16.2 12.1 39.2 39.2 169.85. HPF designThe HPF kinematic and stiffness models were used to design a prototype moving groove HPF. Thecomponents of the prototype are shown in Fig. 16. The structural components and flexures are madefrom 6061 T6 Aluminum. The groove flexures are equipped with a hardened stainless steel plate, called agroove plate, which has been polished to a mirror finish. The plates are bonded to the dorsal surface ofeach groove flexure. The balls are comprised of hardened stainless steel which has been polished tomirror finish. The balls are bonded and pressed into brass collars which are in turn bonded and pressedinto the top octagonal fixture plate. It is this plate which may be used to fixture and position other partswhich are attached to it. The six guide flexures have been cut into a monolithic base component whichis made from 6061 T6 Aluminum.Figure 16: Prototype moving groove HPF, fixture envelope is 250mmx250mmx80mmThe cross section of the ball-grove joint shown in Fig. 17 illustrates how the grooves, groove flexures andgroove plates are rigidly attached to the monolithic base which contains the guiding flexures. Piezoelectricactuators are integrated in the plane of the compliant mechanism such that they are structurally inparallel with their corresponding flexure bearing.14